LMFDB - Embedded newform 5733.2.a.bh.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 7 $ x^4 - 6*x^2 + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.10100$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10100 q^{2} +2.41421 q^{4} +2.97127 q^{5} +0.870264 q^{8} +O(q^{10})$	$q+2.10100 q^{2} +2.41421 q^{4} +2.97127 q^{5} +0.870264 q^{8} +6.24264 q^{10} +2.97127 q^{11} -1.00000 q^{13} -3.00000 q^{16} +1.74053 q^{17} +7.17327 q^{20} +6.24264 q^{22} +1.74053 q^{23} +3.82843 q^{25} -2.10100 q^{26} +5.94253 q^{29} +5.17157 q^{31} -8.04354 q^{32} +3.65685 q^{34} -6.48528 q^{37} +2.58579 q^{40} +5.43275 q^{41} +1.65685 q^{43} +7.17327 q^{44} +3.65685 q^{46} +9.63475 q^{47} +8.04354 q^{50} -2.41421 q^{52} +8.82843 q^{55} +12.4853 q^{58} -3.69222 q^{59} -3.65685 q^{61} +10.8655 q^{62} -10.8995 q^{64} -2.97127 q^{65} -8.48528 q^{67} +4.20201 q^{68} +5.43275 q^{71} +16.1421 q^{73} -13.6256 q^{74} -6.00000 q^{79} -8.91380 q^{80} +11.4142 q^{82} +7.17327 q^{83} +5.17157 q^{85} +3.48106 q^{86} +2.58579 q^{88} +17.3178 q^{89} +4.20201 q^{92} +20.2426 q^{94} -3.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4}+O(q^{10}) $ 4 * q + 4 * q^4	$ 4 q + 4 q^{4} + 8 q^{10} - 4 q^{13} - 12 q^{16} + 8 q^{22} + 4 q^{25} + 32 q^{31} - 8 q^{34} + 8 q^{37} + 16 q^{40} - 16 q^{43} - 8 q^{46} - 4 q^{52} + 24 q^{55} + 16 q^{58} + 8 q^{61} - 4 q^{64} + 8 q^{73} - 24 q^{79} + 40 q^{82} + 32 q^{85} + 16 q^{88} + 64 q^{94} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^10 - 4 * q^13 - 12 * q^16 + 8 * q^22 + 4 * q^25 + 32 * q^31 - 8 * q^34 + 8 * q^37 + 16 * q^40 - 16 * q^43 - 8 * q^46 - 4 * q^52 + 24 * q^55 + 16 * q^58 + 8 * q^61 - 4 * q^64 + 8 * q^73 - 24 * q^79 + 40 * q^82 + 32 * q^85 + 16 * q^88 + 64 * q^94 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.10100	1.48563	0.742817	−	0.669495i	$-0.233489\pi$
$2$	2.10100	1.48563	0.742817	+	0.669495i	$0.233489\pi$
$3$	0	0
$3$	0	0
$4$	2.41421	1.20711
$5$	2.97127	1.32879	0.664395	−	0.747381i	$-0.268689\pi$
$5$	2.97127	1.32879	0.664395	+	0.747381i	$0.268689\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0.870264	0.307685
$9$	0	0
$10$	6.24264	1.97410
$11$	2.97127	0.895871	0.447935	−	0.894066i	$-0.352159\pi$
$11$	2.97127	0.895871	0.447935	+	0.894066i	$0.352159\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	−3.00000	−0.750000
$17$	1.74053	0.422140	0.211070	−	0.977471i	$-0.432305\pi$
$17$	1.74053	0.422140	0.211070	+	0.977471i	$0.432305\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	7.17327	1.60399
$21$	0	0
$22$	6.24264	1.33094
$23$	1.74053	0.362925	0.181463	−	0.983398i	$-0.441917\pi$
$23$	1.74053	0.362925	0.181463	+	0.983398i	$0.441917\pi$
$24$	0	0
$25$	3.82843	0.765685
$26$	−2.10100	−0.412041
$27$	0	0
$28$	0	0
$29$	5.94253	1.10350	0.551750	−	0.834009i	$-0.313960\pi$
$29$	5.94253	1.10350	0.551750	+	0.834009i	$0.313960\pi$
$30$	0	0
$31$	5.17157	0.928842	0.464421	−	0.885615i	$-0.346262\pi$
$31$	5.17157	0.928842	0.464421	+	0.885615i	$0.346262\pi$
$32$	−8.04354	−1.42191
$33$	0	0
$34$	3.65685	0.627145
$35$	0	0
$36$	0	0
$37$	−6.48528	−1.06617	−0.533087	−	0.846061i	$-0.678968\pi$
$37$	−6.48528	−1.06617	−0.533087	+	0.846061i	$0.678968\pi$
$38$	0	0
$39$	0	0
$40$	2.58579	0.408849
$41$	5.43275	0.848452	0.424226	−	0.905556i	$-0.360546\pi$
$41$	5.43275	0.848452	0.424226	+	0.905556i	$0.360546\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	7.17327	1.08141
$45$	0	0
$46$	3.65685	0.539174
$47$	9.63475	1.40537	0.702686	−	0.711500i	$-0.251984\pi$
$47$	9.63475	1.40537	0.702686	+	0.711500i	$0.251984\pi$
$48$	0	0
$49$	0	0
$50$	8.04354	1.13753
$51$	0	0
$52$	−2.41421	−0.334791
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	8.82843	1.19042
$56$	0	0
$57$	0	0
$58$	12.4853	1.63940
$59$	−3.69222	−0.480686	−0.240343	−	0.970688i	$-0.577260\pi$
$59$	−3.69222	−0.480686	−0.240343	+	0.970688i	$0.577260\pi$
$60$	0	0
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	10.8655	1.37992
$63$	0	0
$64$	−10.8995	−1.36244
$65$	−2.97127	−0.368540
$66$	0	0
$67$	−8.48528	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$68$	4.20201	0.509568
$69$	0	0
$70$	0	0
$71$	5.43275	0.644748	0.322374	−	0.946612i	$-0.395519\pi$
$71$	5.43275	0.644748	0.322374	+	0.946612i	$0.395519\pi$
$72$	0	0
$73$	16.1421	1.88929	0.944647	−	0.328088i	$-0.106404\pi$
$73$	16.1421	1.88929	0.944647	+	0.328088i	$0.106404\pi$
$74$	−13.6256	−1.58394
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−8.91380	−0.996593
$81$	0	0
$82$	11.4142	1.26049
$83$	7.17327	0.787369	0.393684	−	0.919246i	$-0.371200\pi$
$83$	7.17327	0.787369	0.393684	+	0.919246i	$0.371200\pi$
$84$	0	0
$85$	5.17157	0.560936
$86$	3.48106	0.375372
$87$	0	0
$88$	2.58579	0.275646
$89$	17.3178	1.83568	0.917842	−	0.396945i	$-0.129930\pi$
$89$	17.3178	1.83568	0.917842	+	0.396945i	$0.129930\pi$
$90$	0	0
$91$	0	0
$92$	4.20201	0.438089
$93$	0	0
$94$	20.2426	2.08787
$95$	0	0
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	0	0
$100$	9.24264	0.924264
$101$	−16.0871	−1.60072	−0.800362	−	0.599517i	$-0.795359\pi$
$101$	−16.0871	−1.60072	−0.800362	+	0.599517i	$0.795359\pi$
$102$	0	0
$103$	1.51472	0.149250	0.0746248	−	0.997212i	$-0.476224\pi$
$103$	1.51472	0.149250	0.0746248	+	0.997212i	$0.476224\pi$
$104$	−0.870264	−0.0853364
$105$	0	0
$106$	0	0
$107$	1.74053	0.168263	0.0841316	−	0.996455i	$-0.473188\pi$
$107$	1.74053	0.168263	0.0841316	+	0.996455i	$0.473188\pi$
$108$	0	0
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	18.5486	1.76854
$111$	0	0
$112$	0	0
$113$	−10.8655	−1.02214	−0.511070	−	0.859539i	$-0.670751\pi$
$113$	−10.8655	−1.02214	−0.511070	+	0.859539i	$0.670751\pi$
$114$	0	0
$115$	5.17157	0.482252
$116$	14.3465	1.33204
$117$	0	0
$118$	−7.75736	−0.714123
$119$	0	0
$120$	0	0
$121$	−2.17157	−0.197416
$122$	−7.68306	−0.695592
$123$	0	0
$124$	12.4853	1.12121
$125$	−3.48106	−0.311355
$126$	0	0
$127$	−15.6569	−1.38932	−0.694661	−	0.719338i	$-0.744445\pi$
$127$	−15.6569	−1.38932	−0.694661	+	0.719338i	$0.744445\pi$
$128$	−6.81280	−0.602172
$129$	0	0
$130$	−6.24264	−0.547516
$131$	−9.42359	−0.823343	−0.411671	−	0.911332i	$-0.635055\pi$
$131$	−9.42359	−0.823343	−0.411671	+	0.911332i	$0.635055\pi$
$132$	0	0
$133$	0	0
$134$	−17.8276	−1.54007
$135$	0	0
$136$	1.51472	0.129886
$137$	10.6543	0.910261	0.455130	−	0.890425i	$-0.349593\pi$
$137$	10.6543	0.910261	0.455130	+	0.890425i	$0.349593\pi$
$138$	0	0
$139$	16.1421	1.36916	0.684579	−	0.728939i	$-0.259986\pi$
$139$	16.1421	1.36916	0.684579	+	0.728939i	$0.259986\pi$
$140$	0	0
$141$	0	0
$142$	11.4142	0.957860
$143$	−2.97127	−0.248470
$144$	0	0
$145$	17.6569	1.46632
$146$	33.9147	2.80680
$147$	0	0
$148$	−15.6569	−1.28699
$149$	3.69222	0.302478	0.151239	−	0.988497i	$-0.451674\pi$
$149$	3.69222	0.302478	0.151239	+	0.988497i	$0.451674\pi$
$150$	0	0
$151$	−16.4853	−1.34155	−0.670777	−	0.741659i	$-0.734039\pi$
$151$	−16.4853	−1.34155	−0.670777	+	0.741659i	$0.734039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.3661	1.23424
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−12.6060	−1.00288
$159$	0	0
$160$	−23.8995	−1.88942
$161$	0	0
$162$	0	0
$163$	−9.65685	−0.756383	−0.378192	−	0.925727i	$-0.623454\pi$
$163$	−9.65685	−0.756383	−0.378192	+	0.925727i	$0.623454\pi$
$164$	13.1158	1.02417
$165$	0	0
$166$	15.0711	1.16974
$167$	3.69222	0.285712	0.142856	−	0.989743i	$-0.454371\pi$
$167$	3.69222	0.285712	0.142856	+	0.989743i	$0.454371\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	10.8655	0.833345
$171$	0	0
$172$	4.00000	0.304997
$173$	21.0100	1.59736	0.798681	−	0.601754i	$-0.205531\pi$
$173$	21.0100	1.59736	0.798681	+	0.601754i	$0.205531\pi$
$174$	0	0
$175$	0	0
$176$	−8.91380	−0.671903
$177$	0	0
$178$	36.3848	2.72715
$179$	19.5681	1.46259	0.731295	−	0.682061i	$-0.238916\pi$
$179$	19.5681	1.46259	0.731295	+	0.682061i	$0.238916\pi$
$180$	0	0
$181$	7.31371	0.543624	0.271812	−	0.962350i	$-0.412377\pi$
$181$	7.31371	0.543624	0.271812	+	0.962350i	$0.412377\pi$
$182$	0	0
$183$	0	0
$184$	1.51472	0.111667
$185$	−19.2695	−1.41672
$186$	0	0
$187$	5.17157	0.378183
$188$	23.2603	1.69644
$189$	0	0
$190$	0	0
$191$	−12.6060	−0.912139	−0.456070	−	0.889944i	$-0.650743\pi$
$191$	−12.6060	−0.912139	−0.456070	+	0.889944i	$0.650743\pi$
$192$	0	0
$193$	−12.8284	−0.923410	−0.461705	−	0.887033i	$-0.652762\pi$
$193$	−12.8284	−0.923410	−0.461705	+	0.887033i	$0.652762\pi$
$194$	−7.68306	−0.551612
$195$	0	0
$196$	0	0
$197$	0.211161	0.0150446	0.00752231	−	0.999972i	$-0.497606\pi$
$197$	0.211161	0.0150446	0.00752231	+	0.999972i	$0.497606\pi$
$198$	0	0
$199$	−24.9706	−1.77012	−0.885058	−	0.465481i	$-0.845882\pi$
$199$	−24.9706	−1.77012	−0.885058	+	0.465481i	$0.845882\pi$
$200$	3.33174	0.235590
$201$	0	0
$202$	−33.7990	−2.37809
$203$	0	0
$204$	0	0
$205$	16.1421	1.12742
$206$	3.18243	0.221730
$207$	0	0
$208$	3.00000	0.208013
$209$	0	0
$210$	0	0
$211$	22.9706	1.58136	0.790679	−	0.612230i	$-0.209728\pi$
$211$	22.9706	1.58136	0.790679	+	0.612230i	$0.209728\pi$
$212$	0	0
$213$	0	0
$214$	3.65685	0.249977
$215$	4.92296	0.335743
$216$	0	0
$217$	0	0
$218$	−16.0871	−1.08955
$219$	0	0
$220$	21.3137	1.43697
$221$	−1.74053	−0.117081
$222$	0	0
$223$	−24.9706	−1.67215	−0.836076	−	0.548613i	$-0.815156\pi$
$223$	−24.9706	−1.67215	−0.836076	+	0.548613i	$0.815156\pi$
$224$	0	0
$225$	0	0
$226$	−22.8284	−1.51852
$227$	−4.71179	−0.312733	−0.156366	−	0.987699i	$-0.549978\pi$
$227$	−4.71179	−0.312733	−0.156366	+	0.987699i	$0.549978\pi$
$228$	0	0
$229$	1.51472	0.100095	0.0500477	−	0.998747i	$-0.484063\pi$
$229$	1.51472	0.100095	0.0500477	+	0.998747i	$0.484063\pi$
$230$	10.8655	0.716449
$231$	0	0
$232$	5.17157	0.339530
$233$	−21.3087	−1.39598	−0.697988	−	0.716109i	$-0.745921\pi$
$233$	−21.3087	−1.39598	−0.697988	+	0.716109i	$0.745921\pi$
$234$	0	0
$235$	28.6274	1.86745
$236$	−8.91380	−0.580239
$237$	0	0
$238$	0	0
$239$	−5.43275	−0.351415	−0.175708	−	0.984442i	$-0.556221\pi$
$239$	−5.43275	−0.351415	−0.175708	+	0.984442i	$0.556221\pi$
$240$	0	0
$241$	−23.4558	−1.51092	−0.755462	−	0.655193i	$-0.772587\pi$
$241$	−23.4558	−1.51092	−0.755462	+	0.655193i	$0.772587\pi$
$242$	−4.56248	−0.293287
$243$	0	0
$244$	−8.82843	−0.565182
$245$	0	0
$246$	0	0
$247$	0	0
$248$	4.50063	0.285791
$249$	0	0
$250$	−7.31371	−0.462560
$251$	−17.8276	−1.12527	−0.562634	−	0.826706i	$-0.690212\pi$
$251$	−17.8276	−1.12527	−0.562634	+	0.826706i	$0.690212\pi$
$252$	0	0
$253$	5.17157	0.325134
$254$	−32.8951	−2.06402
$255$	0	0
$256$	7.48528	0.467830
$257$	−12.6060	−0.786342	−0.393171	−	0.919465i	$-0.628622\pi$
$257$	−12.6060	−0.786342	−0.393171	+	0.919465i	$0.628622\pi$
$258$	0	0
$259$	0	0
$260$	−7.17327	−0.444868
$261$	0	0
$262$	−19.7990	−1.22319
$263$	−5.22158	−0.321977	−0.160988	−	0.986956i	$-0.551468\pi$
$263$	−5.22158	−0.321977	−0.160988	+	0.986956i	$0.551468\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−20.4853	−1.25134
$269$	−12.6060	−0.768602	−0.384301	−	0.923208i	$-0.625558\pi$
$269$	−12.6060	−0.768602	−0.384301	+	0.923208i	$0.625558\pi$
$270$	0	0
$271$	22.8284	1.38673	0.693364	−	0.720587i	$-0.256128\pi$
$271$	22.8284	1.38673	0.693364	+	0.720587i	$0.256128\pi$
$272$	−5.22158	−0.316605
$273$	0	0
$274$	22.3848	1.35231
$275$	11.3753	0.685955
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	33.9147	2.03407
$279$	0	0
$280$	0	0
$281$	3.69222	0.220259	0.110130	−	0.993917i	$-0.464873\pi$
$281$	3.69222	0.220259	0.110130	+	0.993917i	$0.464873\pi$
$282$	0	0
$283$	−7.31371	−0.434755	−0.217377	−	0.976088i	$-0.569750\pi$
$283$	−7.31371	−0.434755	−0.217377	+	0.976088i	$0.569750\pi$
$284$	13.1158	0.778280
$285$	0	0
$286$	−6.24264	−0.369135
$287$	0	0
$288$	0	0
$289$	−13.9706	−0.821798
$290$	37.0971	2.17842
$291$	0	0
$292$	38.9706	2.28058
$293$	1.52937	0.0893465	0.0446733	−	0.999002i	$-0.485775\pi$
$293$	1.52937	0.0893465	0.0446733	+	0.999002i	$0.485775\pi$
$294$	0	0
$295$	−10.9706	−0.638731
$296$	−5.64391	−0.328045
$297$	0	0
$298$	7.75736	0.449372
$299$	−1.74053	−0.100657
$300$	0	0
$301$	0	0
$302$	−34.6356	−1.99306
$303$	0	0
$304$	0	0
$305$	−10.8655	−0.622156
$306$	0	0
$307$	24.9706	1.42515	0.712573	−	0.701598i	$-0.247530\pi$
$307$	24.9706	1.42515	0.712573	+	0.701598i	$0.247530\pi$
$308$	0	0
$309$	0	0
$310$	32.2843	1.83362
$311$	1.01958	0.0578149	0.0289075	−	0.999582i	$-0.490797\pi$
$311$	1.01958	0.0578149	0.0289075	+	0.999582i	$0.490797\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	−14.4853	−0.814861
$317$	−28.4819	−1.59970	−0.799852	−	0.600197i	$-0.795089\pi$
$317$	−28.4819	−1.59970	−0.799852	+	0.600197i	$0.795089\pi$
$318$	0	0
$319$	17.6569	0.988594
$320$	−32.3853	−1.81039
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3.82843	−0.212363
$326$	−20.2891	−1.12371
$327$	0	0
$328$	4.72792	0.261056
$329$	0	0
$330$	0	0
$331$	2.34315	0.128791	0.0643955	−	0.997924i	$-0.479488\pi$
$331$	2.34315	0.128791	0.0643955	+	0.997924i	$0.479488\pi$
$332$	17.3178	0.950438
$333$	0	0
$334$	7.75736	0.424464
$335$	−25.2120	−1.37748
$336$	0	0
$337$	21.4558	1.16877	0.584387	−	0.811475i	$-0.301335\pi$
$337$	21.4558	1.16877	0.584387	+	0.811475i	$0.301335\pi$
$338$	2.10100	0.114279
$339$	0	0
$340$	12.4853	0.677109
$341$	15.3661	0.832122
$342$	0	0
$343$	0	0
$344$	1.44190	0.0777421
$345$	0	0
$346$	44.1421	2.37310
$347$	21.0100	1.12788	0.563939	−	0.825817i	$-0.309285\pi$
$347$	21.0100	1.12788	0.563939	+	0.825817i	$0.309285\pi$
$348$	0	0
$349$	−21.3137	−1.14090	−0.570448	−	0.821333i	$-0.693231\pi$
$349$	−21.3137	−1.14090	−0.570448	+	0.821333i	$0.693231\pi$
$350$	0	0
$351$	0	0
$352$	−23.8995	−1.27385
$353$	−5.43275	−0.289156	−0.144578	−	0.989493i	$-0.546182\pi$
$353$	−5.43275	−0.289156	−0.144578	+	0.989493i	$0.546182\pi$
$354$	0	0
$355$	16.1421	0.856736
$356$	41.8089	2.21587
$357$	0	0
$358$	41.1127	2.17287
$359$	26.7414	1.41136	0.705679	−	0.708532i	$-0.250642\pi$
$359$	26.7414	1.41136	0.705679	+	0.708532i	$0.250642\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	15.3661	0.807626
$363$	0	0
$364$	0	0
$365$	47.9626	2.51048
$366$	0	0
$367$	23.4558	1.22439	0.612193	−	0.790709i	$-0.290288\pi$
$367$	23.4558	1.22439	0.612193	+	0.790709i	$0.290288\pi$
$368$	−5.22158	−0.272194
$369$	0	0
$370$	−40.4853	−2.10473
$371$	0	0
$372$	0	0
$373$	−4.48528	−0.232239	−0.116120	−	0.993235i	$-0.537046\pi$
$373$	−4.48528	−0.232239	−0.116120	+	0.993235i	$0.537046\pi$
$374$	10.8655	0.561841
$375$	0	0
$376$	8.38478	0.432412
$377$	−5.94253	−0.306056
$378$	0	0
$379$	−28.4853	−1.46319	−0.731595	−	0.681739i	$-0.761224\pi$
$379$	−28.4853	−1.46319	−0.731595	+	0.681739i	$0.761224\pi$
$380$	0	0
$381$	0	0
$382$	−26.4853	−1.35510
$383$	5.73137	0.292859	0.146430	−	0.989221i	$-0.453222\pi$
$383$	5.73137	0.292859	0.146430	+	0.989221i	$0.453222\pi$
$384$	0	0
$385$	0	0
$386$	−26.9526	−1.37185
$387$	0	0
$388$	−8.82843	−0.448195
$389$	−11.8851	−0.602597	−0.301298	−	0.953530i	$-0.597420\pi$
$389$	−11.8851	−0.602597	−0.301298	+	0.953530i	$0.597420\pi$
$390$	0	0
$391$	3.02944	0.153205
$392$	0	0
$393$	0	0
$394$	0.443651	0.0223508
$395$	−17.8276	−0.897004
$396$	0	0
$397$	36.8284	1.84837	0.924183	−	0.381950i	$-0.124747\pi$
$397$	36.8284	1.84837	0.924183	+	0.381950i	$0.124747\pi$
$398$	−52.4632	−2.62974
$399$	0	0
$400$	−11.4853	−0.574264
$401$	−4.71179	−0.235296	−0.117648	−	0.993055i	$-0.537535\pi$
$401$	−4.71179	−0.235296	−0.117648	+	0.993055i	$0.537535\pi$
$402$	0	0
$403$	−5.17157	−0.257614
$404$	−38.8376	−1.93224
$405$	0	0
$406$	0	0
$407$	−19.2695	−0.955154
$408$	0	0
$409$	−38.9706	−1.92697	−0.963485	−	0.267762i	$-0.913716\pi$
$409$	−38.9706	−1.92697	−0.963485	+	0.267762i	$0.913716\pi$
$410$	33.9147	1.67493
$411$	0	0
$412$	3.65685	0.180160
$413$	0	0
$414$	0	0
$415$	21.3137	1.04625
$416$	8.04354	0.394367
$417$	0	0
$418$	0	0
$419$	4.50063	0.219870	0.109935	−	0.993939i	$-0.464936\pi$
$419$	4.50063	0.219870	0.109935	+	0.993939i	$0.464936\pi$
$420$	0	0
$421$	−10.4853	−0.511021	−0.255511	−	0.966806i	$-0.582244\pi$
$421$	−10.4853	−0.511021	−0.255511	+	0.966806i	$0.582244\pi$
$422$	48.2612	2.34932
$423$	0	0
$424$	0	0
$425$	6.66348	0.323226
$426$	0	0
$427$	0	0
$428$	4.20201	0.203112
$429$	0	0
$430$	10.3431	0.498791
$431$	26.7414	1.28809	0.644044	−	0.764988i	$-0.277255\pi$
$431$	26.7414	1.28809	0.644044	+	0.764988i	$0.277255\pi$
$432$	0	0
$433$	−10.9706	−0.527212	−0.263606	−	0.964630i	$-0.584912\pi$
$433$	−10.9706	−0.527212	−0.263606	+	0.964630i	$0.584912\pi$
$434$	0	0
$435$	0	0
$436$	−18.4853	−0.885284
$437$	0	0
$438$	0	0
$439$	−23.4558	−1.11949	−0.559743	−	0.828666i	$-0.689100\pi$
$439$	−23.4558	−1.11949	−0.559743	+	0.828666i	$0.689100\pi$
$440$	7.68306	0.366276
$441$	0	0
$442$	−3.65685	−0.173939
$443$	28.3945	1.34906	0.674531	−	0.738247i	$-0.264346\pi$
$443$	28.3945	1.34906	0.674531	+	0.738247i	$0.264346\pi$
$444$	0	0
$445$	51.4558	2.43924
$446$	−52.4632	−2.48421
$447$	0	0
$448$	0	0
$449$	−11.0767	−0.522740	−0.261370	−	0.965239i	$-0.584174\pi$
$449$	−11.0767	−0.522740	−0.261370	+	0.965239i	$0.584174\pi$
$450$	0	0
$451$	16.1421	0.760103
$452$	−26.2316	−1.23383
$453$	0	0
$454$	−9.89949	−0.464606
$455$	0	0
$456$	0	0
$457$	30.9706	1.44874	0.724371	−	0.689410i	$-0.242130\pi$
$457$	30.9706	1.44874	0.724371	+	0.689410i	$0.242130\pi$
$458$	3.18243	0.148705
$459$	0	0
$460$	12.4853	0.582129
$461$	−7.89422	−0.367671	−0.183835	−	0.982957i	$-0.558851\pi$
$461$	−7.89422	−0.367671	−0.183835	+	0.982957i	$0.558851\pi$
$462$	0	0
$463$	3.51472	0.163343	0.0816714	−	0.996659i	$-0.473974\pi$
$463$	3.51472	0.163343	0.0816714	+	0.996659i	$0.473974\pi$
$464$	−17.8276	−0.827626
$465$	0	0
$466$	−44.7696	−2.07391
$467$	−5.94253	−0.274988	−0.137494	−	0.990503i	$-0.543905\pi$
$467$	−5.94253	−0.274988	−0.137494	+	0.990503i	$0.543905\pi$
$468$	0	0
$469$	0	0
$470$	60.1463	2.77434
$471$	0	0
$472$	−3.21320	−0.147900
$473$	4.92296	0.226358
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−11.4142	−0.522074
$479$	−9.63475	−0.440223	−0.220112	−	0.975475i	$-0.570642\pi$
$479$	−9.63475	−0.440223	−0.220112	+	0.975475i	$0.570642\pi$
$480$	0	0
$481$	6.48528	0.295703
$482$	−49.2808	−2.24468
$483$	0	0
$484$	−5.24264	−0.238302
$485$	−10.8655	−0.493376
$486$	0	0
$487$	−16.4853	−0.747019	−0.373510	−	0.927626i	$-0.621846\pi$
$487$	−16.4853	−0.747019	−0.373510	+	0.927626i	$0.621846\pi$
$488$	−3.18243	−0.144062
$489$	0	0
$490$	0	0
$491$	−26.9526	−1.21635	−0.608176	−	0.793802i	$-0.708099\pi$
$491$	−26.9526	−1.21635	−0.608176	+	0.793802i	$0.708099\pi$
$492$	0	0
$493$	10.3431	0.465832
$494$	0	0
$495$	0	0
$496$	−15.5147	−0.696631
$497$	0	0
$498$	0	0
$499$	24.2843	1.08711	0.543557	−	0.839372i	$-0.317077\pi$
$499$	24.2843	1.08711	0.543557	+	0.839372i	$0.317077\pi$
$500$	−8.40401	−0.375839
$501$	0	0
$502$	−37.4558	−1.67174
$503$	−39.5586	−1.76383	−0.881915	−	0.471408i	$-0.843746\pi$
$503$	−39.5586	−1.76383	−0.881915	+	0.471408i	$0.843746\pi$
$504$	0	0
$505$	−47.7990	−2.12703
$506$	10.8655	0.483030
$507$	0	0
$508$	−37.7990	−1.67706
$509$	−6.87465	−0.304713	−0.152357	−	0.988326i	$-0.548686\pi$
$509$	−6.87465	−0.304713	−0.152357	+	0.988326i	$0.548686\pi$
$510$	0	0
$511$	0	0
$512$	29.3522	1.29720
$513$	0	0
$514$	−26.4853	−1.16822
$515$	4.50063	0.198322
$516$	0	0
$517$	28.6274	1.25903
$518$	0	0
$519$	0	0
$520$	−2.58579	−0.113394
$521$	0.720950	0.0315854	0.0157927	−	0.999875i	$-0.494973\pi$
$521$	0.720950	0.0315854	0.0157927	+	0.999875i	$0.494973\pi$
$522$	0	0
$523$	26.4853	1.15812	0.579060	−	0.815285i	$-0.303420\pi$
$523$	26.4853	1.15812	0.579060	+	0.815285i	$0.303420\pi$
$524$	−22.7506	−0.993863
$525$	0	0
$526$	−10.9706	−0.478339
$527$	9.00127	0.392101
$528$	0	0
$529$	−19.9706	−0.868285
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.43275	−0.235318
$534$	0	0
$535$	5.17157	0.223587
$536$	−7.38443	−0.318959
$537$	0	0
$538$	−26.4853	−1.14186
$539$	0	0
$540$	0	0
$541$	31.4558	1.35239	0.676196	−	0.736722i	$-0.263627\pi$
$541$	31.4558	1.35239	0.676196	+	0.736722i	$0.263627\pi$
$542$	47.9626	2.06017
$543$	0	0
$544$	−14.0000	−0.600245
$545$	−22.7506	−0.974527
$546$	0	0
$547$	8.68629	0.371399	0.185700	−	0.982607i	$-0.440545\pi$
$547$	8.68629	0.371399	0.185700	+	0.982607i	$0.440545\pi$
$548$	25.7218	1.09878
$549$	0	0
$550$	23.8995	1.01908
$551$	0	0
$552$	0	0
$553$	0	0
$554$	37.8181	1.60674
$555$	0	0
$556$	38.9706	1.65272
$557$	−29.9238	−1.26791	−0.633957	−	0.773369i	$-0.718570\pi$
$557$	−29.9238	−1.26791	−0.633957	+	0.773369i	$0.718570\pi$
$558$	0	0
$559$	−1.65685	−0.0700775
$560$	0	0
$561$	0	0
$562$	7.75736	0.327224
$563$	−42.0201	−1.77093	−0.885467	−	0.464702i	$-0.846161\pi$
$563$	−42.0201	−1.77093	−0.885467	+	0.464702i	$0.846161\pi$
$564$	0	0
$565$	−32.2843	−1.35821
$566$	−15.3661	−0.645886
$567$	0	0
$568$	4.72792	0.198379
$569$	−15.7884	−0.661886	−0.330943	−	0.943651i	$-0.607367\pi$
$569$	−15.7884	−0.661886	−0.330943	+	0.943651i	$0.607367\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−7.17327	−0.299930
$573$	0	0
$574$	0	0
$575$	6.66348	0.277886
$576$	0	0
$577$	−11.8579	−0.493649	−0.246825	−	0.969060i	$-0.579387\pi$
$577$	−11.8579	−0.493649	−0.246825	+	0.969060i	$0.579387\pi$
$578$	−29.3522	−1.22089
$579$	0	0
$580$	42.6274	1.77001
$581$	0	0
$582$	0	0
$583$	0	0
$584$	14.0479	0.581307
$585$	0	0
$586$	3.21320	0.132736
$587$	−35.8664	−1.48036	−0.740182	−	0.672407i	$-0.765261\pi$
$587$	−35.8664	−1.48036	−0.740182	+	0.672407i	$0.765261\pi$
$588$	0	0
$589$	0	0
$590$	−23.0492	−0.948920
$591$	0	0
$592$	19.4558	0.799630
$593$	−33.7035	−1.38404	−0.692019	−	0.721880i	$-0.743278\pi$
$593$	−33.7035	−1.38404	−0.692019	+	0.721880i	$0.743278\pi$
$594$	0	0
$595$	0	0
$596$	8.91380	0.365124
$597$	0	0
$598$	−3.65685	−0.149540
$599$	−38.8376	−1.58686	−0.793431	−	0.608660i	$-0.791707\pi$
$599$	−38.8376	−1.58686	−0.793431	+	0.608660i	$0.791707\pi$
$600$	0	0
$601$	10.9706	0.447499	0.223749	−	0.974647i	$-0.428170\pi$
$601$	10.9706	0.447499	0.223749	+	0.974647i	$0.428170\pi$
$602$	0	0
$603$	0	0
$604$	−39.7990	−1.61940
$605$	−6.45232	−0.262324
$606$	0	0
$607$	−19.1716	−0.778150	−0.389075	−	0.921206i	$-0.627205\pi$
$607$	−19.1716	−0.778150	−0.389075	+	0.921206i	$0.627205\pi$
$608$	0	0
$609$	0	0
$610$	−22.8284	−0.924296
$611$	−9.63475	−0.389780
$612$	0	0
$613$	−14.2843	−0.576936	−0.288468	−	0.957489i	$-0.593146\pi$
$613$	−14.2843	−0.576936	−0.288468	+	0.957489i	$0.593146\pi$
$614$	52.4632	2.11724
$615$	0	0
$616$	0	0
$617$	38.3278	1.54302	0.771510	−	0.636217i	$-0.219502\pi$
$617$	38.3278	1.54302	0.771510	+	0.636217i	$0.219502\pi$
$618$	0	0
$619$	30.1421	1.21151	0.605757	−	0.795649i	$-0.292870\pi$
$619$	30.1421	1.21151	0.605757	+	0.795649i	$0.292870\pi$
$620$	37.0971	1.48986
$621$	0	0
$622$	2.14214	0.0858918
$623$	0	0
$624$	0	0
$625$	−29.4853	−1.17941
$626$	29.4140	1.17562
$627$	0	0
$628$	0	0
$629$	−11.2878	−0.450075
$630$	0	0
$631$	48.2843	1.92217	0.961083	−	0.276259i	$-0.0890947\pi$
$631$	48.2843	1.92217	0.961083	+	0.276259i	$0.0890947\pi$
$632$	−5.22158	−0.207703
$633$	0	0
$634$	−59.8406	−2.37657
$635$	−46.5207	−1.84612
$636$	0	0
$637$	0	0
$638$	37.0971	1.46869
$639$	0	0
$640$	−20.2426	−0.800161
$641$	−27.2512	−1.07636	−0.538179	−	0.842831i	$-0.680887\pi$
$641$	−27.2512	−1.07636	−0.538179	+	0.842831i	$0.680887\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.1741	1.26490	0.632448	−	0.774603i	$-0.282050\pi$
$647$	32.1741	1.26490	0.632448	+	0.774603i	$0.282050\pi$
$648$	0	0
$649$	−10.9706	−0.430632
$650$	−8.04354	−0.315493
$651$	0	0
$652$	−23.3137	−0.913035
$653$	−14.3465	−0.561424	−0.280712	−	0.959792i	$-0.590571\pi$
$653$	−14.3465	−0.561424	−0.280712	+	0.959792i	$0.590571\pi$
$654$	0	0
$655$	−28.0000	−1.09405
$656$	−16.2982	−0.636339
$657$	0	0
$658$	0	0
$659$	−29.4140	−1.14581	−0.572904	−	0.819622i	$-0.694183\pi$
$659$	−29.4140	−1.14581	−0.572904	+	0.819622i	$0.694183\pi$
$660$	0	0
$661$	3.65685	0.142235	0.0711176	−	0.997468i	$-0.477343\pi$
$661$	3.65685	0.142235	0.0711176	+	0.997468i	$0.477343\pi$
$662$	4.92296	0.191336
$663$	0	0
$664$	6.24264	0.242261
$665$	0	0
$666$	0	0
$667$	10.3431	0.400488
$668$	8.91380	0.344885
$669$	0	0
$670$	−52.9706	−2.04643
$671$	−10.8655	−0.419458
$672$	0	0
$673$	−26.9706	−1.03964	−0.519819	−	0.854276i	$-0.674001\pi$
$673$	−26.9706	−1.03964	−0.519819	+	0.854276i	$0.674001\pi$
$674$	45.0788	1.73637
$675$	0	0
$676$	2.41421	0.0928544
$677$	−1.74053	−0.0668939	−0.0334470	−	0.999440i	$-0.510648\pi$
$677$	−1.74053	−0.0668939	−0.0334470	+	0.999440i	$0.510648\pi$
$678$	0	0
$679$	0	0
$680$	4.50063	0.172591
$681$	0	0
$682$	32.2843	1.23623
$683$	30.6448	1.17259	0.586295	−	0.810098i	$-0.300586\pi$
$683$	30.6448	1.17259	0.586295	+	0.810098i	$0.300586\pi$
$684$	0	0
$685$	31.6569	1.20955
$686$	0	0
$687$	0	0
$688$	−4.97056	−0.189501
$689$	0	0
$690$	0	0
$691$	12.4853	0.474962	0.237481	−	0.971392i	$-0.423678\pi$
$691$	12.4853	0.474962	0.237481	+	0.971392i	$0.423678\pi$
$692$	50.7227	1.92819
$693$	0	0
$694$	44.1421	1.67561
$695$	47.9626	1.81932
$696$	0	0
$697$	9.45584	0.358166
$698$	−44.7802	−1.69495
$699$	0	0
$700$	0	0
$701$	−8.40401	−0.317415	−0.158708	−	0.987326i	$-0.550733\pi$
$701$	−8.40401	−0.317415	−0.158708	+	0.987326i	$0.550733\pi$
$702$	0	0
$703$	0	0
$704$	−32.3853	−1.22057
$705$	0	0
$706$	−11.4142	−0.429580
$707$	0	0
$708$	0	0
$709$	−32.3431	−1.21467	−0.607336	−	0.794445i	$-0.707762\pi$
$709$	−32.3431	−1.21467	−0.607336	+	0.794445i	$0.707762\pi$
$710$	33.9147	1.27280
$711$	0	0
$712$	15.0711	0.564812
$713$	9.00127	0.337100
$714$	0	0
$715$	−8.82843	−0.330164
$716$	47.2416	1.76550
$717$	0	0
$718$	56.1838	2.09676
$719$	−36.0775	−1.34546	−0.672732	−	0.739886i	$-0.734879\pi$
$719$	−36.0775	−1.34546	−0.672732	+	0.739886i	$0.734879\pi$
$720$	0	0
$721$	0	0
$722$	−39.9191	−1.48563
$723$	0	0
$724$	17.6569	0.656212
$725$	22.7506	0.844935
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	0	0
$730$	100.770	3.72965
$731$	2.88380	0.106661
$732$	0	0
$733$	26.4853	0.978256	0.489128	−	0.872212i	$-0.337315\pi$
$733$	26.4853	0.978256	0.489128	+	0.872212i	$0.337315\pi$
$734$	49.2808	1.81899
$735$	0	0
$736$	−14.0000	−0.516047
$737$	−25.2120	−0.928697
$738$	0	0
$739$	−18.8284	−0.692615	−0.346307	−	0.938121i	$-0.612565\pi$
$739$	−18.8284	−0.692615	−0.346307	+	0.938121i	$0.612565\pi$
$740$	−46.5207	−1.71013
$741$	0	0
$742$	0	0
$743$	29.6252	1.08684	0.543422	−	0.839460i	$-0.317128\pi$
$743$	29.6252	1.08684	0.543422	+	0.839460i	$0.317128\pi$
$744$	0	0
$745$	10.9706	0.401930
$746$	−9.42359	−0.345022
$747$	0	0
$748$	12.4853	0.456507
$749$	0	0
$750$	0	0
$751$	−12.2843	−0.448259	−0.224130	−	0.974559i	$-0.571954\pi$
$751$	−12.2843	−0.448259	−0.224130	+	0.974559i	$0.571954\pi$
$752$	−28.9043	−1.05403
$753$	0	0
$754$	−12.4853	−0.454687
$755$	−48.9822	−1.78264
$756$	0	0
$757$	24.4853	0.889933	0.444966	−	0.895547i	$-0.353216\pi$
$757$	24.4853	0.889933	0.444966	+	0.895547i	$0.353216\pi$
$758$	−59.8477	−2.17376
$759$	0	0
$760$	0	0
$761$	−28.1833	−1.02164	−0.510822	−	0.859687i	$-0.670659\pi$
$761$	−28.1833	−1.02164	−0.510822	+	0.859687i	$0.670659\pi$
$762$	0	0
$763$	0	0
$764$	−30.4336	−1.10105
$765$	0	0
$766$	12.0416	0.435082
$767$	3.69222	0.133318
$768$	0	0
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	0	0
$772$	−30.9706	−1.11465
$773$	−41.0879	−1.47783	−0.738915	−	0.673798i	$-0.764662\pi$
$773$	−41.0879	−1.47783	−0.738915	+	0.673798i	$0.764662\pi$
$774$	0	0
$775$	19.7990	0.711201
$776$	−3.18243	−0.114243
$777$	0	0
$778$	−24.9706	−0.895238
$779$	0	0
$780$	0	0
$781$	16.1421	0.577611
$782$	6.36486	0.227607
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30.1421	1.07445	0.537226	−	0.843439i	$-0.319472\pi$
$787$	30.1421	1.07445	0.537226	+	0.843439i	$0.319472\pi$
$788$	0.509789	0.0181605
$789$	0	0
$790$	−37.4558	−1.33262
$791$	0	0
$792$	0	0
$793$	3.65685	0.129859
$794$	77.3766	2.74599
$795$	0	0
$796$	−60.2843	−2.13672
$797$	−17.1067	−0.605949	−0.302974	−	0.952999i	$-0.597980\pi$
$797$	−17.1067	−0.605949	−0.302974	+	0.952999i	$0.597980\pi$
$798$	0	0
$799$	16.7696	0.593264
$800$	−30.7941	−1.08874
$801$	0	0
$802$	−9.89949	−0.349563
$803$	47.9626	1.69256
$804$	0	0
$805$	0	0
$806$	−10.8655	−0.382721
$807$	0	0
$808$	−14.0000	−0.492518
$809$	−24.7897	−0.871560	−0.435780	−	0.900053i	$-0.643527\pi$
$809$	−24.7897	−0.871560	−0.435780	+	0.900053i	$0.643527\pi$
$810$	0	0
$811$	12.4853	0.438418	0.219209	−	0.975678i	$-0.429652\pi$
$811$	12.4853	0.438418	0.219209	+	0.975678i	$0.429652\pi$
$812$	0	0
$813$	0	0
$814$	−40.4853	−1.41901
$815$	−28.6931	−1.00508
$816$	0	0
$817$	0	0
$818$	−81.8773	−2.86277
$819$	0	0
$820$	38.9706	1.36091
$821$	39.3474	1.37323	0.686617	−	0.727019i	$-0.259095\pi$
$821$	39.3474	1.37323	0.686617	+	0.727019i	$0.259095\pi$
$822$	0	0
$823$	36.2843	1.26479	0.632395	−	0.774646i	$-0.282072\pi$
$823$	36.2843	1.26479	0.632395	+	0.774646i	$0.282072\pi$
$824$	1.31820	0.0459218
$825$	0	0
$826$	0	0
$827$	−37.1846	−1.29303	−0.646517	−	0.762900i	$-0.723775\pi$
$827$	−37.1846	−1.29303	−0.646517	+	0.762900i	$0.723775\pi$
$828$	0	0
$829$	−49.3137	−1.71274	−0.856368	−	0.516366i	$-0.827284\pi$
$829$	−49.3137	−1.71274	−0.856368	+	0.516366i	$0.827284\pi$
$830$	44.7802	1.55434
$831$	0	0
$832$	10.8995	0.377872
$833$	0	0
$834$	0	0
$835$	10.9706	0.379652
$836$	0	0
$837$	0	0
$838$	9.45584	0.326647
$839$	−5.73137	−0.197869	−0.0989345	−	0.995094i	$-0.531543\pi$
$839$	−5.73137	−0.197869	−0.0989345	+	0.995094i	$0.531543\pi$
$840$	0	0
$841$	6.31371	0.217714
$842$	−22.0296	−0.759190
$843$	0	0
$844$	55.4558	1.90887
$845$	2.97127	0.102215
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	14.0000	0.480196
$851$	−11.2878	−0.386941
$852$	0	0
$853$	−23.4558	−0.803113	−0.401556	−	0.915834i	$-0.631531\pi$
$853$	−23.4558	−0.803113	−0.401556	+	0.915834i	$0.631531\pi$
$854$	0	0
$855$	0	0
$856$	1.51472	0.0517720
$857$	55.2233	1.88639	0.943197	−	0.332235i	$-0.107803\pi$
$857$	55.2233	1.88639	0.943197	+	0.332235i	$0.107803\pi$
$858$	0	0
$859$	−32.2843	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$859$	−32.2843	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$860$	11.8851	0.405277
$861$	0	0
$862$	56.1838	1.91363
$863$	44.5690	1.51715	0.758573	−	0.651588i	$-0.225897\pi$
$863$	44.5690	1.51715	0.758573	+	0.651588i	$0.225897\pi$
$864$	0	0
$865$	62.4264	2.12256
$866$	−23.0492	−0.783243
$867$	0	0
$868$	0	0
$869$	−17.8276	−0.604760
$870$	0	0
$871$	8.48528	0.287513
$872$	−6.66348	−0.225654
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	−49.2808	−1.66315
$879$	0	0
$880$	−26.4853	−0.892819
$881$	12.6060	0.424708	0.212354	−	0.977193i	$-0.431887\pi$
$881$	12.6060	0.424708	0.212354	+	0.977193i	$0.431887\pi$
$882$	0	0
$883$	8.34315	0.280769	0.140385	−	0.990097i	$-0.455166\pi$
$883$	8.34315	0.280769	0.140385	+	0.990097i	$0.455166\pi$
$884$	−4.20201	−0.141329
$885$	0	0
$886$	59.6569	2.00421
$887$	13.9242	0.467530	0.233765	−	0.972293i	$-0.424895\pi$
$887$	13.9242	0.467530	0.233765	+	0.972293i	$0.424895\pi$
$888$	0	0
$889$	0	0
$890$	108.109	3.62382
$891$	0	0
$892$	−60.2843	−2.01847
$893$	0	0
$894$	0	0
$895$	58.1421	1.94348
$896$	0	0
$897$	0	0
$898$	−23.2721	−0.776599
$899$	30.7322	1.02498
$900$	0	0
$901$	0	0
$902$	33.9147	1.12924
$903$	0	0
$904$	−9.45584	−0.314497
$905$	21.7310	0.722362
$906$	0	0
$907$	−39.9411	−1.32622	−0.663112	−	0.748520i	$-0.730765\pi$
$907$	−39.9411	−1.32622	−0.663112	+	0.748520i	$0.730765\pi$
$908$	−11.3753	−0.377502
$909$	0	0
$910$	0	0
$911$	59.5490	1.97295	0.986474	−	0.163919i	$-0.0524137\pi$
$911$	59.5490	1.97295	0.986474	+	0.163919i	$0.0524137\pi$
$912$	0	0
$913$	21.3137	0.705381
$914$	65.0692	2.15230
$915$	0	0
$916$	3.65685	0.120826
$917$	0	0
$918$	0	0
$919$	−4.34315	−0.143267	−0.0716336	−	0.997431i	$-0.522821\pi$
$919$	−4.34315	−0.143267	−0.0716336	+	0.997431i	$0.522821\pi$
$920$	4.50063	0.148381
$921$	0	0
$922$	−16.5858	−0.546224
$923$	−5.43275	−0.178821
$924$	0	0
$925$	−24.8284	−0.816354
$926$	7.38443	0.242668
$927$	0	0
$928$	−47.7990	−1.56908
$929$	41.5103	1.36191	0.680954	−	0.732326i	$-0.261565\pi$
$929$	41.5103	1.36191	0.680954	+	0.732326i	$0.261565\pi$
$930$	0	0
$931$	0	0
$932$	−51.4436	−1.68509
$933$	0	0
$934$	−12.4853	−0.408531
$935$	15.3661	0.502526
$936$	0	0
$937$	0.627417	0.0204968	0.0102484	−	0.999947i	$-0.496738\pi$
$937$	0.627417	0.0204968	0.0102484	+	0.999947i	$0.496738\pi$
$938$	0	0
$939$	0	0
$940$	69.1127	2.25421
$941$	−8.91380	−0.290582	−0.145291	−	0.989389i	$-0.546412\pi$
$941$	−8.91380	−0.290582	−0.145291	+	0.989389i	$0.546412\pi$
$942$	0	0
$943$	9.45584	0.307925
$944$	11.0767	0.360514
$945$	0	0
$946$	10.3431	0.336285
$947$	−10.9530	−0.355923	−0.177962	−	0.984037i	$-0.556950\pi$
$947$	−10.9530	−0.355923	−0.177962	+	0.984037i	$0.556950\pi$
$948$	0	0
$949$	−16.1421	−0.523996
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−49.4045	−1.60037	−0.800184	−	0.599754i	$-0.795265\pi$
$953$	−49.4045	−1.60037	−0.800184	+	0.599754i	$0.795265\pi$
$954$	0	0
$955$	−37.4558	−1.21204
$956$	−13.1158	−0.424196
$957$	0	0
$958$	−20.2426	−0.654010
$959$	0	0
$960$	0	0
$961$	−4.25483	−0.137253
$962$	13.6256	0.439307
$963$	0	0
$964$	−56.6274	−1.82385
$965$	−38.1167	−1.22702
$966$	0	0
$967$	−33.9411	−1.09147	−0.545737	−	0.837957i	$-0.683750\pi$
$967$	−33.9411	−1.09147	−0.545737	+	0.837957i	$0.683750\pi$
$968$	−1.88984	−0.0607418
$969$	0	0
$970$	−22.8284	−0.732977
$971$	22.3282	0.716547	0.358274	−	0.933617i	$-0.383366\pi$
$971$	22.3282	0.716547	0.358274	+	0.933617i	$0.383366\pi$
$972$	0	0
$973$	0	0
$974$	−34.6356	−1.10980
$975$	0	0
$976$	10.9706	0.351159
$977$	−28.4819	−0.911218	−0.455609	−	0.890180i	$-0.650578\pi$
$977$	−28.4819	−0.911218	−0.455609	+	0.890180i	$0.650578\pi$
$978$	0	0
$979$	51.4558	1.64454
$980$	0	0
$981$	0	0
$982$	−56.6274	−1.80705
$983$	−46.7319	−1.49051	−0.745257	−	0.666777i	$-0.767673\pi$
$983$	−46.7319	−1.49051	−0.745257	+	0.666777i	$0.767673\pi$
$984$	0	0
$985$	0.627417	0.0199912
$986$	21.7310	0.692055
$987$	0	0
$988$	0	0
$989$	2.88380	0.0916995
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	−41.5977	−1.32073
$993$	0	0
$994$	0	0
$995$	−74.1942	−2.35211
$996$	0	0
$997$	−52.9706	−1.67759	−0.838797	−	0.544444i	$-0.816741\pi$
$997$	−52.9706	−1.67759	−0.838797	+	0.544444i	$0.816741\pi$
$998$	51.0213	1.61505
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bh.1.4	yes	4
3.2	odd	2	inner	5733.2.a.bh.1.1	yes	4
7.6	odd	2		5733.2.a.bg.1.4	yes	4
21.20	even	2		5733.2.a.bg.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5733.2.a.bg.1.1	✓	4	21.20	even	2
5733.2.a.bg.1.4	yes	4	7.6	odd	2
5733.2.a.bh.1.1	yes	4	3.2	odd	2	inner
5733.2.a.bh.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q+2.10100 q^{2} +2.41421 q^{4} +2.97127 q^{5} +0.870264 q^{8} +O(q^{10})\)	\(q+2.10100 q^{2} +2.41421 q^{4} +2.97127 q^{5} +0.870264 q^{8} +6.24264 q^{10} +2.97127 q^{11} -1.00000 q^{13} -3.00000 q^{16} +1.74053 q^{17} +7.17327 q^{20} +6.24264 q^{22} +1.74053 q^{23} +3.82843 q^{25} -2.10100 q^{26} +5.94253 q^{29} +5.17157 q^{31} -8.04354 q^{32} +3.65685 q^{34} -6.48528 q^{37} +2.58579 q^{40} +5.43275 q^{41} +1.65685 q^{43} +7.17327 q^{44} +3.65685 q^{46} +9.63475 q^{47} +8.04354 q^{50} -2.41421 q^{52} +8.82843 q^{55} +12.4853 q^{58} -3.69222 q^{59} -3.65685 q^{61} +10.8655 q^{62} -10.8995 q^{64} -2.97127 q^{65} -8.48528 q^{67} +4.20201 q^{68} +5.43275 q^{71} +16.1421 q^{73} -13.6256 q^{74} -6.00000 q^{79} -8.91380 q^{80} +11.4142 q^{82} +7.17327 q^{83} +5.17157 q^{85} +3.48106 q^{86} +2.58579 q^{88} +17.3178 q^{89} +4.20201 q^{92} +20.2426 q^{94} -3.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4}+O(q^{10}) \) 4 * q + 4 * q^4	\( 4 q + 4 q^{4} + 8 q^{10} - 4 q^{13} - 12 q^{16} + 8 q^{22} + 4 q^{25} + 32 q^{31} - 8 q^{34} + 8 q^{37} + 16 q^{40} - 16 q^{43} - 8 q^{46} - 4 q^{52} + 24 q^{55} + 16 q^{58} + 8 q^{61} - 4 q^{64} + 8 q^{73} - 24 q^{79} + 40 q^{82} + 32 q^{85} + 16 q^{88} + 64 q^{94} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^10 - 4 * q^13 - 12 * q^16 + 8 * q^22 + 4 * q^25 + 32 * q^31 - 8 * q^34 + 8 * q^37 + 16 * q^40 - 16 * q^43 - 8 * q^46 - 4 * q^52 + 24 * q^55 + 16 * q^58 + 8 * q^61 - 4 * q^64 + 8 * q^73 - 24 * q^79 + 40 * q^82 + 32 * q^85 + 16 * q^88 + 64 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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